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with d(t) must be included in the internal model system (assuming
they have not already been included). For example, if the nominal input
supplied to the system differs from the required nominal input by a
constant, the internal model system must contain integrators.
Now consider the case when the nominal state is not generated
correctly. Let us write
xd(t) = x (t) xa(t) (5-16)
where xa(t) is the nominal state which is actually supplied to the
*
system, x (t) is the correct nominal state which should have been
supplied, and xd(t) is the disturbance representing the difference
between the correct and actual signals. Since the nominal state is fed
through to the input via a linear feedback gain matrix (see Figure 5.1)
it is apparent that xd(t) can be modeled as an input disturbance.
Hence, we may conclude that the dynamics associated with xd(t) must
also be included in the internal model system.
To summarize, we have shown that robustness with respect to the
ic *
open-loop signals x (t) and u (t) is obtained provided that any
deviations from these signals are sucessfully modeled in the dynamics of
the internal model system.
Digital Implementation
In the previous treatment of the servomechanism problem there has
been an underlying assumption that the control will be implemented via
continuous-time methods. Often it is desirable to implement the control
using a digital computer and hence a discrete-time control law is